The graph of a system of linear equations is shown in the xy-plane. If \(\mathrm{(x, y)}\) is the solution to...

1. TRANSLATE the problem information

• The problem asks us to find the value of \(\mathrm{x + 2y}\)
• We need to first find \(\mathrm{(x, y)}\), which is the solution to the system of linear equations shown
• Key insight: The solution to a system is where the two lines cross

2. TRANSLATE the intersection point from the graph

• Locate where the solid line and dashed line intersect
• To find the x-coordinate: Look straight down from the intersection point to the x-axis
  • The intersection is at \(\mathrm{x = -3}\)
• To find the y-coordinate: Look straight across from the intersection point to the y-axis
  • The intersection is at \(\mathrm{y = 2}\)
• Therefore, the solution is \(\mathrm{(x, y) = (-3, 2)}\)
• Note: The graph legend confirms this: "Intersection: (-3, 2)"

3. SIMPLIFY by substituting and evaluating

• Substitute \(\mathrm{x = -3}\) and \(\mathrm{y = 2}\) into the expression \(\mathrm{x + 2y}\):
  • \(\mathrm{x + 2y = (-3) + 2(2)}\)
• Follow order of operations (multiply first):
  • \(\mathrm{= (-3) + 4}\)
• Add the numbers:
  • \(\mathrm{= 1}\)

Answer: B (1)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students mix up the x and y coordinates when reading from the graph

Many students correctly locate the intersection point but then reverse the coordinates, reading it as \(\mathrm{(2, -3)}\) instead of \(\mathrm{(-3, 2)}\). This happens because they don't systematically look down to the x-axis for the x-coordinate and across to the y-axis for the y-coordinate.

If they use \(\mathrm{(2, -3)}\):

\(\mathrm{x + 2y = (2) + 2(-3)}\)

\(\mathrm{= 2 + (-6)}\)

\(\mathrm{= 2 - 6}\)

\(\mathrm{= -4}\)

This doesn't match any answer choice exactly, leading to confusion and guessing. However, if they make additional errors in calculation, they might arrive at Choice A (-1).

Second Most Common Error:

Poor SIMPLIFY execution: Students make arithmetic errors with negative numbers

Some students correctly identify the intersection as \(\mathrm{(-3, 2)}\) but then make calculation errors:

• Forgetting the negative sign: treating \(\mathrm{-3}\) as \(\mathrm{3}\), giving \(\mathrm{3 + 4 = 7}\) (not an option)
• Forgetting to multiply: calculating \(\mathrm{-3 + 2 + 2 = 1}\) (correct by accident)
• Sign errors: calculating \(\mathrm{-3 + 2 × 2 = -3 - 4 = -7}\) (treating multiplication result as negative)

These arithmetic mistakes can lead to selecting Choice A (-1) or cause confusion that leads to guessing.

The Bottom Line:

This problem tests your ability to accurately read graphical information and work carefully with negative numbers. The key is to be systematic: identify the intersection point, carefully read both coordinates with attention to signs, and methodically apply order of operations when evaluating the expression.